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Definite integrals from area formulas

Problem 4.1064 · easy Video

Evaluate \( \displaystyle \int_{-1}^{1} - x - 1 \, dx \) using a formula from geometry.

The region between the graph and the x-axis is a trapezoid (a triangle, since one side has height 0), below the axis.
Reviewed
\[ \frac{\left(-2 + 0\right) \left(1 + 1\right)}{2} = -2 \]
Its (signed) area.✓ Proved
\[ \int\limits_{-1}^{1} \left(- x - 1\right)\, dx = -2 \]
Integrating directly gives the same.✓ Proved

Answer \( -2 \)

The graph of f(x) = -x - 1 from x = -1 to x = 1. Rectangles under the curve get thinner until they fill the area, -2. Area below the x-axis counts as negative.

Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	numerical quadrature

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the region as a trapezoid (specifically a triangle) below the x-axis and applies the geometric area formula with the correct sign for the definite integral.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the region as a trapezoid (specifically a triangle) below the x-axis and applies the geometric area formula with the correct sign for the definite integral.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the region as a trapezoid (specifically a triangle) below the x-axis and correctly calculates the signed area as -2. The geometric interpretation and calculation are sound.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-14

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/integral_geometry, checked 2026-10-17 with SymPy 1.14.0.